This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384374 #7 May 28 2025 01:09:25
%S A384374 0,0,0,0,0,30,390,3080,19236,104874,524250,2471172,11176968,49065302,
%T A384374 210698670,890007456,3712887756,15342622434,62938446690,256735466012,
%U A384374 1042705518960,4220535078990,17038468898550,68644258099320,276111986410740
%N A384374 Expansion of e.g.f. exp(x)*(exp(x) - 1)*(exp(x) - x - 1)^2.
%C A384374 a(n) is the number of strings of length n defined on {0, 1, 2, 3} with at least one 1, at least two 2's, at least two 3's, and with no restriction on the number of 0's.
%H A384374 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (19,-156,726,-2109,3963,-4814,3644,-1560,288).
%F A384374 a(n) = 4^n - (2*n + 9)*3^(n-1) + (n^2 + 7*n + 12)*2^(n-2) - (n^2 + n + 1).
%F A384374 G.f.: x^5*(30 - 180*x + 350*x^2 - 224*x^3)/((1 - 4*x)*(1 - 3*x)^2*(1 - 3*x + 2*x^2)^3). - _Stefano Spezia_, May 27 2025
%e A384374 a(6) = 390 since the strings are the 180 permutations of 012233, the 90 permutations of 112233, the 60 permutations of 122233, and the 60 permutations of 122333.
%Y A384374 Cf. A000478.
%K A384374 nonn,easy
%O A384374 0,6
%A A384374 _Enrique Navarrete_, May 27 2025